Illiquidity Fire-sales and Capital Structure

Illiquidity, Fire-sales and CapitalStructure∗

Afrasiab MirzaDepartment of Economics

Queen’s [email protected]

November 22, 2011

Abstract

This paper investigates the industry dynamics and capital structure of firms that

hold illiquid assets and face potential fire-sales. Firms enter the industry by paying

a fixed cost financed by equity and debt. Once in operation, shareholders have the

option to adjust their asset holdings at a cost but also retain the option to exit the

industry by defaulting on their debt. While costly, asset sales allow the shareholders

to boost dividends or to service debt payments rather than defaulting. However, as

shareholders make their decisions once debt is in place, the resulting conflict with

bondholders entails over-investment and early liquidation due to debt-overhang. A

substantial number of firms also exit following an exogenous financial shock. In the

stationary industry equilibrium, firms selling assets but are not defaulting find it more

costly to reduce capacity due to the price effects of fire-sale liquidations. This price

feedback effect results in lower industry leverage but a higher default rate. Capital

regulation reduces leverage ex-ante but at the cost of inducing more default. Restrict-

ing asset sales mitigates fire-sales and also reduce leverage ex-ante.

JEL Classification Codes: G28, G32, G33

Keywords: Capital Structure, Fire-Sales, Illiquidity, Macro-prudential Regulation

∗I thank my advisors Thor Koeppl, Frank Milne and Jan Zabojnik for their patience and numer-ous valuable suggestions and comments. I would also like to thank seminar participants at Queen’s.This research has been supported by the Ontario Graduate Scholarship (OGS) awards program,and The John Deutsch Institute (JDI). All omissions and errors are my own.

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1 Introduction

The recent crisis has called into question the effectiveness of regulation forfinancial institutions. These regulations were designed to limit the probabilityof failure of individual institutions. They require problem institutions to eithersell declining assets or raise equity. However, the indifference shown to themethod of adjustment may be problematic when a large number of financialinstitutions are in difficulty as this may precipitate excessive balance-sheetshrinkage leading to a credit crunch.

This has led to an alternative approach to regulation, termed “macro-prudential” regulation, aimed at preventing these types of problems. Hanson,Kashyap, and Stein (2010) define this approach “as an effort to control thesocial costs associated with excessive balance-sheet shrinkage on the part ofmultiple financial institutions hit with a common shock.” They further identifytwo costs of generalized balance-sheet shrinkage: credit-crunches and fire-sales.When financial intermediaries reduce their assets by making reductions innew lending to operating firms, the result is a reduction in “investment andemployment with contractionary consequences for the macroeconomy.” Fire-sales, the simultaneous attempt by many firms to shed assets, may lead to largedeclines in prices as described by Shleifer and Vishny (1992, 2011). These twoeffects are closed linked since a decline in asset prices may necessitate furtherasset reductions, leading to further declines in prices.

The challenge is to understand why financial institutions don’t take ade-quate steps to mitigate these effects. Hanson, Kashyap, and Stein (2010) pointto the debt-overhang problem as the reason financial institutions are incapableof raising fresh equity once a crisis is underway. Moreover, financial institu-tions also fail to take adequate steps prior to a crisis when there is a preferencefor debt. The reason is that they don’t take into account the negative priceexternality their fire-sales impose on the value of collateral of other institutionsduring a crisis. Stein (2010) offers an account along these lines.

In this paper, I extend the framework of Stein (2010) to analyze the dy-namic capital structure choice of financial intermediaries in an industry wherefirms compete in an a market for financial intermediation services but also in amarket for capital or assets that make these services feasible. Firms enter theindustry by purchasing an initial stock of assets at a fixed cost financed by eq-uity and debt. Once in operation, shareholders have the option to adjust theirasset holdings at a cost but also retain the option to exit the industry by de-faulting on their debt. While costly, capacity reductions allow the shareholdersto boost dividends or to service debt payments rather than defaulting. How-ever, as shareholders make their decisions once debt is in place, the resultingconflict with bondholders entails over-investment and early liquidation due to

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debt-overhang. Moreover, following Hanson, Kashyap, and Stein (2010), thispaper proscribes a role for equity injections by outsiders. Nevertheless, capitalinjections from existing shareholders are permitted and default only resultswhen shareholders choose to forgo such injections in favour of liquidation.

Meanwhile, firms also face the prospect of immediate liquidation, to beinterpreted as the realization of an adverse financial shock. At any givenmoment, a significant number of firms exit the industry and liquidate theirasset holdings in the process. The paper examines the impact potential fire-sales have on the determination of ex-ante capital structure and also the impacton the default decision of firms not directly impacted by the financial shock. Inthe stationary industry equilibrium, firms selling assets that are not defaultingfind it more costly to reduce capacity due to the price effects of liquidations bydefaulting firms. This price feedback effect results in lower industry leveragebut a higher default rate.

In terms of the implications for regulation, the findings here stress theimportance of preventing disorderly liquidation and the need for reducingleverage ex-ante. While reducing leverage, capital regulations appear counter-productive as they tend to reinforce the debt-overhang problem. Here, therequirement to maintain the value of equity at or above a pre-determinedlevel is equivalent to a fixed cost, much like the cost of debt service. Thisleads shareholders to over-invest in unproductive assets and exit earlier, butas bondholders anticipate this, leverage is lower.

This paper is related to the large literature on dynamic contingent analysisbeginning with Black and Scholes (1973) and Merton (1974). Specifically, itdraws on the real options valuation literature. Brennan and Schwartz (1984),Mello and Parsons (1992) as well as Titman and Tsyplakov (2007) analyzethe interaction between financing decisions and default through the use ofnumerical methods. Dixit (1992) and Caballero (1991) as well as Bertolaand Caballero (1994) study industry equilibria of all equity financed firmsthat make entry and exit decisions. Leland (1994, 1998), and Morellec (2001)analyze the optimal capital structure choices of a single firm. Miao (2005)studies investment and optimal capital structure capital decisions are costlessto reverse.

This paper also relates to the literature on macro-prudential regulation. Inan early contribution, Blum and Hellwig (1995) show that rigid capital ade-quacy regulation for banks may reinforce macroeconomic fluctuations. Repulloand Suarez (2010) offer important modifications to current capital regulationsthat help mitigate the severity of credit crunches following an aggregate shock.Hanson, Kashyap, and Stein (2010) analyze qualitatively the effectiveness ofseveral proposed macro-prudential regulatory measures. Their analysis under-lines the need to proceed cautiously in raising capital requirements. However,

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they emphasize the need to reduce leverage ex-ante through the imposition ofhair-cuts on asset purchases. The findings here largely support these conclu-sions.

This paper is organized as follows. The following section details the mainelements of the model. Section 3 analyzes the investment and optimal capitalstructures choices of a firm that may only reduce capacity at a cost. Section4 extends these results to also permit costly capacity expansions. Then, inSection 5, the industry equilibrium is analyzed, prior to concluding.

2 Model

I consider a financial intermediation industry with a large number of fi-nancial intermediaries or firms. All investors and firms are risk-neutral anddiscount future cash flows at a constant risk-free rate ρ > 0. Time varies con-tinuously over [0,∞). All stochastic processes are defined over the probabilityspace (Ω,F ,P) which describes the uncertainty in the economy.

2.1 Financial Intermediaries

There are a continuum of ex-ante identical firms. At each instant, everyfirm holds a stocks of assets a that generate instantaneous operating profitgiven by:

π(x, a) = pxa1−γ − cfwhere γ ∈ (0, 1), cf > 0, p is the price of intermediation services prevailingin the market, and x represents firm-specific asset productivity. In aboveexpression, xaγ is the quantity of intermediation services provided so thatpxaγ are the flow revenues, while cf is a fixed operating cost. In addition,xtt≥0 follows a geometric Brownian motion

dxt/xt = µxdt+ σxdWt (1)

where µx, σx > 0 and Wtt≥0 is a standard Brownian motion representingfirm-specific shifts in productivity. Growth in firm-specific productivity cap-tures financial innovation that expands the intermediation capabilities of thefirm.

Financial intermediaries often adjust their asset stocks in response to mar-ket conditions.1 Here, the firms can reduce their intermediation capacity by

1Adrian and Shin (2010) document changes in asset levels and leverage for large financialintermediaries.

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selling assets (disinvesting) to outsiders at a price q per unit. I consider firstthe case where firms can only sell assets so that sales are effectively irreversiblereductions in capacity as in Morellec (2001). Then, in Section 4, I relax thisassumption by allowing firms to also expand capacity through additional assetpurchases.

2.2 Intermediation and Resale Markets

Firms compete in an intermediation market, with the quantity of assetsheld by a firm determining the level of intermediation provided by the firm.Competition entails that individual firms act as price-takers in the intermedi-ation market. The price that each firm faces at every instant t is then givenby:

p = D(Y ) (2)

where D(Y ) is the inverse aggregate demand for intermediation services. D(Y )is a decreasing function of the aggregate quantity of intermediation servicesY . For tractability, I assume that D(Y ) is of the following functional form:

D(Y ) = Y −1ε (3)

where ε is the elasticity of demand.Firms can sell units of their assets to outsiders. The price prevailing at any

time in the resale market is a function of aggregate assets Al for sale at thattime, and individual firms act as price-takers in the resale market. Assumingan aggregate demand function of the iso-elastic form in the resale market, theresale price q is given by:

q = A− 1ε′

l (4)

where ε′ is the elasticity of demand in the resale market. While the assetsacross various firms need not be identical, I assume they are treated as suchby outside investors as the latter are insufficiently informed to distinguishthem.

2.3 Unlevered Firm

The unlevered firm’s objective is to choose an investment policy and aban-donment rule to maximize the expected value of the stream of discountedprofits. The instantaneous profits of the firm are:

π(x, a; p) = pxa1−γ − cf (5)

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where it takes the intermediation price p as given.Now, let ut denote cumulative gross disinvestment (i.e. sales of assets) up

to date t. The stochastic process for assets is:

dat = −dut. (6)

Then, given an intermediation price p and a resale price q, the unlevered firmstarting with an initial stock a and demand x solves the following problem:

vu(x, a; p, q)

= maxT∈T ,ut,t∈[0,T ]

(1− τ)Ex[∫ T

0e−(ρ+η)t π(xt, at; p)dt+ qdut

](7)

where ut0≤t≤T is a nondecreasing continuous stochastic process with u0 = 0,T is the set of all stopping times relative to the filtration generated by theBrownian motion Wtt≥0, Ex is the expectation taken with respect to theprocess xtt≥0 when the starting value is x, τ is the tax rate on corporateincome, and η is a Poisson death rate. The parameter η captures shock to thefinancial system affected a positive measure of firms that subsequently liquidatetheir asset holdings and exit. This exogenous exit channel exacerbates sellingpressure in the market for assets resulting in equilibrium prices below theirfundamental value. I interpret the sales of these firms as “fire-sales.”

The abandonment decision is an option to abandon that is exercised thefirst time asset productivity falls below a threshold level xu(au; p, q), whereau is the asset stock at abandonment. The investment decision correspondsto a continuum of options that are exercised the first time asset productivityfalls below a threshold curve xl(au; p, q) for a > au. This curves traces outthe combinations of asset productivity, x and stock a that equate the marginalbenefit from selling to the marginal product of assets in place.

2.4 Debt and Liquidation Value

Firms issue debt because interest payments to debt are tax deductible.Debt is issued at par with infinite maturity following Leland (1994) and Duffieand Lando (2001). The firm is obligated by the debt contract to pay a couponb to bondholders as long as it is in operation. The residual profit-flow isdistributed among shareholders. Upon default the firm is liquidated imme-diately,2 at which time the bondholders receive the liquidation value and theshareholders are wiped out.

2The underlying assumption here is that debt restructuring is very costly. If one considers thecase of large financial intermediaries, the The failures of Bear Stearns and Lehman Brothers provide

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Mello and Parsons (1992), Morellec (2001) and Miao (2005), model liquida-tion value as a fraction of the unlevered firm value vu(x, a; p, q). The unleveredfirm value is equal to the after-tax present value of the flow of profits, plusthe value associated with the options to alter asset levels and the option toabandon the assets. Financial intermediaries typically own assets that aremarked-to-market. The liquidation value of a financial intermediary is thentypically the market value of its asset holding. For this reason, I specify theabandonment value of the unlevered firm to be the market value of its assets,namely qa if the firm has a assets on hand at the time of abandonment.

2.5 Equity, Investment and Liquidation

Given a coupon b, the shareholders make investment and liquidation deci-sions to maximize the value of equity:

e(x, a, b; p, q)

= maxT∈T ,ut,t∈[0,T ]

(1− τ)Ex[∫ T

0e−(ρ+η)t [π(xt, at; p)− b]dt+ qdut

]. (8)

Note that the value of equity is increasing in the resale price q. Moreover,costs of debt service operate in the same manner as the fixed costs cf .

The default decision corresponds to an option to abandon which is exercisedthe first time demand falls below a threshold level xd(ad, b; p, q) that dependson the coupon b, and where ad is the asset stock at default. The investmentdecision again corresponds to a continuum of options to reduce capacity. Theseoptions are exercised whenever asset productivity falls below the thresholdcurve xl(a; p, q) as long as a > ad. Notice that as the coupon essentiallyimposes a fixed cost on shareholders, it does not impact the selling thresholdxl as the latter is pinned down by the marginal costs and benefits of reducingcapacity.

2.6 Debt Value

Debt holders are entitled to the coupon payments b while the firm is inoperation along with the abandonment value. Bankruptcy is costly and thusthe abandonment value is a is a fraction (1 − ζ) ∈ (0, 1) of the market valueqad of the firm’s assets at default. Hence, the arbitrage-free value of debt

examples of costly liquidations in the case of financial intermediaries.

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d(x, a, b; p, q) is given by:

d(x, a, b; p, q) = Ex[∫ Txd

0e−(ρ+η)tbdt

]+ (1− ζ)qadE

x[e−(ρ+η)Txd ] (9)

where Txd denotes the first time firm-specific demand falls below the defaultthreshold xd.

2.7 Firm Value

The value of the firm v(x, a, b; p, q) is the sum of the value of the equityand debt. Hence,

v(x, a, b; p, q) = e(x, a, b; p, q) + d(x, a, b; p, q) (10)

2.8 Entry

There are a continuum of potential firms that can enter the industry ateach instant by incurring a fixed sunk cost of entry ce. This cost is financed bydebt and equity. The initial productivity and size are drawn uniformly fromthe set [x, x]× [a, a]. The draws across firms are assumed to be independent.Upon entering, firms are not obliged to begin providing intermediation servicesimmediately. Entry is to be viewed as a costly mechanism to observe the initialfirm-specific asset productivity, and acquire a stock of assets. As a result,firms may choose to wait till asset productivity is sufficiently high to beginoperations.

I assume that x > xd(ad, b; p, q) and a > ad, assumptions that will needto be verified in equilibrium as xd and ad are determined endogenously. Asin Miao (2005), these assumptions serve to avoid the situation where firmsenter and exit immediately. After entry, asset productivity of an entrant isgiven by the process xtt≥0 that is identical to existing firms. However, giventhe different starting values of x, firms face different sequences of productivitylevels xtt≥0.

As firms are all identical prior to entering the industry, the ex-ante valueto entering must equal the entry cost in equilibrium. Thus, the free-entrycondition can be written as:∫ x

x

∫ a

av(x, a, b; p, q)F (da× dx) = ce (11)

where F is the uniform distribution on [x, x]× [a, a]. The optimal choice of thecoupon is made by firms prior to entering. They select b∗(a; p, q) to maximize

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the expected value∫ xx

∫ aa v(x, a, b; p, q)F (da × dx). As all firms are ex-ante

identical they choose the same coupon in equilibrium.

2.9 Timing

The timing of the decisions is shown in the figure below:

Entry financing, b chosen

Initial demand Demand follows a GBM

x xt

Default and liquidation

Continue operating

Asset Sales

ut

Figure 1: Timing

2.10 Aggregates

The long-run steady state is characterized by a stationary distribution ofsurviving firms ν, and a constant entry rate N . Note that if ν is stationary, sowill the equilibrium asset holdings as long as the asset levels of entrants canbe chosen appropriately. Then, given a Borel set B in R2, ν(B) is the numberof surviving firms with (x, a) in B. The support of ν is [xd,∞) × [ad,∞).Moreover, we can compute aggregates as follows:

Y =

∫ x

xd

∫ a

ad

[xa1−γ ]ν(dx× da) (12)

Al =

∫ xl

xd

∫ a

ad

[a− x−1l (xl(a; p, q)− x; p, q)]ν(dx× da) (13)

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where x−1(·) is the inverse of the selling threshold function xl. It is importantto emphasize that ν is not a probability distribution.

2.11 Equilibrium

In the case where only capacity contractions are permitted, a stationary in-dustry equilibrium is a constant intermediation price p∗, a constant resale priceq∗, an exit/default asset stock a∗d, an exit/default threshold xd(a

∗d, b∗; p∗, q∗),

an entry rate N∗, and a stationary distributions ν∗ such that:

(i) b∗ is determined via arg maxb∫ xx

∫ aa v(x, a, b; p, q)F (da× dx)

(ii) shareholders solve (8)

(iii) markets clear:

p∗ = Y (ν∗, a∗d, b∗)−

1ε (14)

q∗ = Al(ν∗, a∗d, b

∗)−1ε′ (15)

(iv) the free-entry condition (11) holds

(v) the distribution ν∗ is an invariant measures over [x∗d,∞)× [a∗d,∞)

The stationarity of the equilibrium distributions implies that the distributionof firms is constant over time. Implicitly, due to the idiosyncratic nature ofshocks a law of large numbers is assumed. Nevertheless, considerable dynamicsunderlie this stationary equilibrium. At any point in time, a large numberof firms are entering while an equal amount are exiting. Similarly, a largenumber of firms are selling assets, either to service debt payments or becauseof liquidations following default, while a large number of firms are holdingcapacity fixed. Finally, the above definition of equilibrium can naturally bemodified to apply when capacity expansions are also permitted.

3 Optimal Capital Structure

In this section, I solve the model where only capacity reductions are possi-ble. This case is instructive as closed-form expressions for the optimal aban-donment and investment decisions, along with firm value can be obtained.These results closely mirror those in Morellec (2001). In the next section,these results are extended to the case when capacity expansions are also pos-sible.

Shareholders make investment and default decisions after debt is in placeand are protected by limited liability, so they seek to expropriate bondholders.

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The latter anticipate this and as a result, the optimal capital structure isinefficient. The resulting debt-overhang problem leads shareholders to over-invest in unproductive assets and exit early, as in Myers (1977). Moreover,debt-overhang causes increased delays in investing as

3.1 Unlevered Firm

As noted earlier, the objective of shareholders in the unlevered firm isto make abandonment and investment decisions that maximize the expectedstream of discounted profits. Formally, the problem is as follows:

vu(x, a; p, q)

= maxT∈T ,ut,t∈[0,T ]

(1− τ)Ex[∫ T

0e−(ρ+η)t π(xt, at; p)dt+ qdut

]Before presenting a formal solution to this problem, I proceed heuristicallyto derive the optimal decisions and firm value relying on intuition from thereal-options literature. This approach makes the economic tradeoffs involvedin the decisions more transparent.

The value of the unlevered firm stems from three sources. The first sourceof value is the right to the discounted profits from operating the firm’s assetsforever. The second source of value comes from the continuum of irreversibleoptions to sell assets, or reduce capacity. Finally, the third source of valuederives from the value associated with the (irreversible) option to default.Thus, to determine the value of the firm, it suffices to sum the value of eachof these options.

The value from operating the firm’s assets forever without altering capacitywhen the initial asset level is a and initial asset productivity is x can be writtenas:

Π(x, a; p) = (1− τ)Ex[∫ ∞

0e−(ρ+η)t π(xt, at; p)dt

].

This is of course just the value of the discounted stream of profits in perpe-tuity. Given that instantaneous profits are linear in asset productivity, theexpectation above can be explicitly computed so that Π(x, a; p) can be writtenas

(1− τ)

(pxa1−γ

(ρ+ η)− µx− c

ρ+ η

). (16)

This expression is simply the difference between the after-tax present-valuesof revenues and costs.

The value from capacity reductions can be viewed as a continuum of op-

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tions available to the firm. The firm starts with an initial stock of assets a,and can reduce capacity to every level below a. Each possible reduction con-stitutes an option, and since there are continuum of possible reductions, thefirm holds a continuum of options. The optimal capacity reduction decisiontrades off the marginal product of assets against the sale price. Variability inasset productivity naturally leads to changes in the marginal product of assets.When the marginal product falls below the resale price, assets are sold.

As the costs of adjusting capacity are linear, the optimal investment policyis defined by a continuous increasing threshold function xl(a; p, q). Thus, givenan initial asset level a, if productivity falls below xl(a; p, q) assets are sold. Theproceeds from these sales are paid out as dividends to shareholders. The firmundertakes these reductions in capacity because they boost dividends whenasset productivity is low. On the other hand, if asset productivity is abovexl(a; p, q), the firm does nothing, as the option to sell is not “in the money.”Following Morellec (2001), the value of the option to reduce capacity can bewritten as:∫ au

a

(qEx[e−(ρ+η)Txl(a;p,q) ] −

(1− τ)Ex[∫ ∞

Txl(a;p,q)

e−(ρ+η)tπa(xt, at; p)dt

]da

)(17)

where πa(·) is the marginal product of assets, and au is the firm size at aban-donment. In the expression above, the first term captures the expected dis-counted proceeds from sales. The second term captures the loss in profitsresulting from a permanent reduction in capacity. The optimal selling thresh-old is found using the super-contact condition:

∂2vu∂x∂a

∣∣∣x=xl(a;p,q)

= 0 (18)

The option to abandon the firm’s assets is a single irreversible option.The optimal abandonment policy trades off the value of continuing operationsagainst the liquidation value. The former fluctuates due to changes in assetproductivity. The firm then chooses to liquidate when the value generated byassets in place fall below the liquidation value. The firm’s abandonment policyis defined by another continuous threshold function xu(a; p, q) such that givenan asset stock a, the firm liquidates the first time asset productivity falls belowxu(a; p, q). Firm size at abandonment, au, is determined by equating the twothresholds, xl(au; p, q) = xu(au; p, q). That is, the firm is abandoned when theliquidation value of the firm exceeds the value from continuing operations at

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a lower capacity.The value from abandonment is the sum of the liquidation value discounted

till abandonment minus the discounted cash flows from continuing operationsat abandonment:

qauEx[e−(ρ+η)Txu ]− (1− τ)Exu[∫ Txu

0e−(ρ+η)tπ(xt, at; p)dt

].

Exploiting the strong Markov property of stopping times to rewrite the secondterm the value of the abandonment option is:

qauEx[e−(ρ+η)Txu ]− Ex[e−(ρ+η)Txu ]

[∫ ∞0

e−(ρ+η)tπ(xt, at; p)dt

]Then, noting that Ex[e−(ρ+η)Txu ] = (x/xu)−λ, the value is

[qau −Π(xu, au; p)] (x/xu)−λ. (19)

The optimal abandonment threshold xu is found via the following smooth-pasting condition:

∂vu∂x

∣∣∣x=xu

= 0. (20)

The optimal decisions of the firm along with its value are described in thefollowing proposition:

Proposition 1. Assume (ρ+ η) > µx > 0. Denote by −λ be the negative rootof the fundamental quadratic:

(ρ+ η)− µxϑ−1

2σ2(ϑ− 1)ϑ = 0. (21)

Then,

vu = Π(x, a; p) +q

(1 + λ)(1 + γλ)(a(x/xl(a; p, q))−λ − au(x/xl(au; p, q))−λ

+ [qau −Π(xu, au; p)] (x/xu(au; p, q))−λ (22)

where the capacity reducation and abandonment thresholds are:

xl(a; p, q) =λq((ρ+ η)− µx)

(1 + λ)(1− γ)(1− τ)paγ (23)

xu(au; p, q) =λ((ρ+ η)− µx) [(1− τ)(c/(ρ+ η) + qau] aγ−1

u

(1 + λ)(1− τ)p(24)

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Finally, the firm size upon abandonment is:

au =c(1− γ)(1− τ)

q(ρ+ η)γ(25)

Proof. See Appendix A.

In the above expression for the value of the firm, vu, the first term is thevalue from operating the firm’s assets forever, the second corresponds to thevalue from capacity reductions while the final term captures the value fromthe option to abandon. The optimal investment and abandonment policiesare illustrated in the figure below: The selling threshold xl is increasing with

a

x

Xl(a;p,q)

Xu(a;p,q)

au

Xu(au;p,q)

Sell Assets

Do Nothing

Abandon Abandon

Abandon

Figure 2: The optimal investment and abandonment policies.

a. Above xl, no action is taken as capacity expansions are not possible, whilebelow xl capacity reductions are undertaken. On the other hand, the abandon-ment threshold xu is decreasing in a. Below this threshold the liquidation valueof the firm exceeds the value from continuing to operate. The intersection ofthe two thresholds pins down the size upon abandonment au.

The above expressions allow us to deduce an important difference in howchanges in the intermediation price p and the resale price q affect the value

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of the firm. First consider changes in the intermediation price p. Clearly,increasing p raises the value from operating the assets, Π(x, a; p). However,by substituting the expression for xl in vu, it is clear that capacity reductionsare less valuable as p is raised. This is because the marginal product of assetsgoes up as p increases, reducing the need for reducing capacity. Similarly,abandonment is less valuable as p is raised because the foregone profits fromcontinuing operations are higher. The overall effects from changes in p on thevalue of the firm are ambiguous because increases in operating profits are offsetby declines in the values of the options.

Now consider the effect of changing the resale price q. Clearly, changingq does not affect the profits from operating the firm’s assets in perpetuity,Π(x, a; p). However, raising q increases the value from capacity reductions.This is because a higher q results in more proceeds from asset sales. Similarly,the value of the option to default (given by the last term in vu) is also increasingwith the resale price as increasing q raises the liquidation value qau. Thus,while increasing p decreases the values of the options available to the firm,increasing q raises them. The overall effects from changes in q on value areunambiguous: raising q increases the value of the firm.

Table 3.1 below summarizes the effects discussed in the above, along witha few additional comparative statics. The base parameters were chosen tobe comparable to those of Morellec (2001) and Miao (2005) as follows: γ =0.53, a = 100, x = 1, ρ = 0.05, η = 0.01, σx = 0.1, τ = 0.15, c = 1, p = 1, q =1, η = 0.01.

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Comparative StaticsParameter Value Value Value of Size

Value without capacity with capacity selling uponchanges changes option (%) default ad

Baseline 133.9 144.5 7.8% 12.6

γ = 0.5 155.8 160.5 3.0% 14.2γ = 0.6 93.1 163.4 75.3% 9.4

σx = 0.05 133.9 160.5 19.9% 12.6σx = 0.15 132.9 145.2 9.3% 12.6

p = 0.5 - 238.3 - 12.6p = 1.5 - 209.9 - 12.6

q = 0.5 - 134.0 - 25.1q = 1.5 - 218.0 - 8.4

τ = 0.05 149.7 156.2 4.3% 14.0τ = 0.55 70.9 212.8 200.1% 6.7

The results show that firm value is no longer monotonically increasing inp. For instance, when p is increased from 0.5 to 1, value drops from 238.3to 144.5, whereas the value then jumps back up to 209.9 as p is raised to1.5. By contrast, increases in the resale price result in significant increases invalue (nearly 50% compared with the baseline), and also lead to much lowerabandonment levels.

The effect of changes in the tax rate on firm value are also not monotone.In fact, firm value does increase as taxes as lowered, however it also jumps upfor very high tax rates as capacity reductions and abandonment become muchmore valuable. This can be seen in the last line of the table above.

It is also interesting to note that changes in the volatility in asset produc-tivity do not affect firm value monotonically. This is because more volatilityraises the value from the ability to reduce capacity. Indeed, the results aboveshow an increase in value from either increasing or decreasing volatility.

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3.2 Levered Firm

The firm has a preference for issuing debt over raising equity because ofthe inherent tax advantages of debt. I consider infinite maturity debt con-tracts that consist of a coupon b to be paid to bondholders in perpetuity alongwith a commitment to liquidate the firm upon default. Moreover, absolutepriority is enforced in that the proceeds from liquidation are used to first paybondholders before paying shareholders. Default is triggered when sharehold-ers choose to abandon the firm’s assets rather than inject new capital to meetdebt obligations.

Formally, given a coupon b shareholders solve the following problem:

e(x, a, b; p, q)

= maxT∈T ,ut,t∈[0,T ]

(1− τ)Ex[∫ T

0e−(ρ+η)t [π(xt, at; p)− b]dt+ qdut

]When managers operate the firm in the interest of shareholders, they makecapacity and default decisions to maximize the value of equity. Equity is valu-able because shareholders have residual income rights along with the optionto alter the firm’s capacity, and the option to default. The option to reducecapacity by selling assets is valuable for two reasons. Selling assets when theyare less productive allows the firm to either pay additional dividends or meetdebt obligations without raising additional equity.

Following the same real-options approach to compute the value of equity,the value to shareholders arising from their right to operate the firm’s assetsforever is:

Π(x, a, b; p) = (1− τ)Ex[∫ ∞

0e−(ρ+η)t [π(xt, at; p)− b]dt

](26)

As is clear from the above expression, the cost of servicing debt reduces thediscounted flow of profits available to shareholders.

The option to default trades off the value from continuing operations againstthe liquidation value. Debt obligations reduce the profits from operations whilebankruptcy costs reduce the liquidation value. When the reduction in prof-its dominates the reduction in liquidation value, shareholders default earlierthan in the unlevered firm. Essentially, there is inefficient liquidation due todebt-overhang as in Myers (1977).

The value to shareholders from the options to reduce capacity is similarto the value of these options in the unlevered case. This is because the fixedcoupon b does not alter either the marginal costs of reducing capacity nor the

17

resale price q. Hence, the value of the options to reduce capacity is:∫ ad

a

(qEx[e−(ρ+η)Txl(a;p,q) ] −

(1− τ)Ex[∫ ∞

Txl(a;p,q)

e−(ρ+η)tπa(xt, at; p)dt

]da

). (27)

where ad is the size of the firm upon default. Again, the optimal capacityreduction threshold is found via the super-contact condition:

∂2vu∂x∂a

∣∣∣x=xl(a;p,q)

= 0 (28)

The value of equity is then the sum of the values described above. Formally,the problem is similar to the problem of the unlevered firm and is solved usingthe same method. The optimal policies chosen by shareholders and the valueof equity are summarized in proposition below:

Proposition 2. Assume (ρ+ η) > µx > 0. Again denote by −λ the negativeroot of the fundamental quadratic:

(ρ+ η)− µxϑ−1

2σ2(ϑ− 1)ϑ = 0. (29)

Then,

e(x, a, b; p, q) = Π(x, a, b; p)+q

(1 + λ)(1 + γλ)(a(x/xl(a; p, q))−λ−ad(x/xl(ad; p, q))−λ

−Π(xd, ad, b; p)

(x

xd

)−λ(30)

where the capacity reduction and default thresholds are:

xl(a; p, q) =λqaγ((ρ+ η)− µx)

p(1− γ)(1 + λ)(1− τ)(31)

xd(a, b; p, q) =λaγ−1((ρ+ η)− µx)(c+ b)

p(ρ+ η)(1 + λ)(32)

Finally, the firm size upon default is:

ad =(c+ b)(1− γ)(1− τ)

q(ρ+ η)(33)

Proof. See Appendix A.

18

Corollary 1. When debt service costs are sufficiently higher than the fixedcosts, i.e. c < γb

1−γ , then ad > au.

The expression for equity above differs from the unlevered value of the firmin two aspects. When maximizing the value of equity with debt in place, share-holders choose a default threshold xd rather than xu. In addition, shareholdersdo not receive the liquidation value qad upon default.

Most importantly, the above results show that the conflict between share-holders and bondholders leads to debt-overhang. To see this, first note that byexamining (32) and (33), it is clear that both the default threshold xd(a, b; p, q)and the firm size upon default ad are both increasing in the coupon b. In otherwords, higher leverage increases the probability of default and investment inunproductive assets. Then, when the fixed operating cost is sufficiently low,higher leverage induces shareholders to exit early and over-invest in unproduc-tive assets relative to the unlevered firm.

The optimal decisions of shareholders are depicted below:

a

x

Xl(a;p,q)

Xu(a;p,q)

ad

Xd(au;p,q)

Sell Assets

Do Nothing

Abandon Abandon

Abandon

Xd(a,b;p,q)

Figure 3: Optimal policies chosen by shareholders.

19

The figure clearly exhibits the debt-overhang problem: the default thresholdcurve is above the abandonment threshold curve of the unlevered firm. Thisimplies a higher probability of default or early exit. Furthermore, as the exitsize ad is higher than the unlevered abandonment size au, shareholders over-invest in unproductive assets than are abandoned upon default.

Given the optimal decisions of shareholders, the value of debt is:

d(x, a, b; p, q) = Ex[∫ Txd

0e−(ρ+η)tbdt

]+ (1− ζ)qadE

x[e−(ρ+η)Txd ] (34)

=b

(ρ+ η)

(1−

(x

xd

)−λ)+ (1− ζ)qad

(x

xd

)−λ. (35)

As the above expression makes clear, the value of debt is essentially the dis-counted value of the coupon b/(ρ + η) till default plus the liquidation valuediscounted appropriately.

The value of the levered firm can then be written simply as the sum of thevalues of equity and debt:

Proposition 3. The value of the levered firm is:

v(x, a, b; p, q)

= Π(x, a, b; p) +q

(1 + λ)(1 + γλ)(a(x/xl(a; p, q))−λ − ad(x/xl(ad; p, q))−λ

+ [(1− ζ)qad −Π(xd, ad, b; p)]

(x

xd

)−λ+

b

(ρ+ η)

(1−

(x

xd

)−λ)(36)

In order to clarify the role of the tax-shield, we can re-write the value of thelevered firm using the expressions for Π(x, a, b; p) and Π(xd, ad, b; p) as follows:

v(x, a, b; p, q)

= Π(x, a; p) +q

(1 + λ)(1 + γλ)(a(x/xl(a; p, q))−λ − ad(x/xl(ad; p, q))−λ

+ [(1− ζ)qad −Π(xd, ad; p)] (x/xd)−λ +

τb

(ρ+ η)

(1−

(x

xd

)−λ)(37)

In comparison with the expression for the value of the unlevered firm, clearlythe default and abandonment thresholds are different. In addition, the lastterm is new. This term captures the value of the tax shield. It is simply equalsthe value of the taxes on expected discounted stream of coupon payments untildefault. Finally, ζ captures the bankruptcy costs.

20

In terms of comparative statics, the effects of changes in p and q are againquite different. Just as in the unlevered case, raising p increases profits fromoperations but also decreases the value from capacity reductions. However,leverage renders the effects of raising p on the value of the default optionambiguous. This is because the increase in profits foregone upon default froma higher p is offset by a lower probability of default. Overall, raising p ismore likely to boost the value of equity when the firm is levered. Moreover,the changes in the value of debt are also ambiguous. This is because raisingp reduces the probability of default and so increases the expected dividendpayments yet reduces the expected liquidation payout.

As for the effects of changes in q, it is clear that the value of equity isincreasing in the resale price it raises the value of capacity reductions whileleaving everything else unaltered. Similarly, the value of debt is also increasingin q as it raises the liquidation value of the firm. Hence, the value of the leveredfirm is monotonically increasing in the resale value q.

Table 3.2 summarizes the comparative statics of the model. The baseparameters were again chosen to be comparable to those of Morellec (2001) asfollows: γ = 0.53, a = 100, x = 1, ρ = 0.05, η = 0.01, σx = 0.1, τ = 0.15, c =1, p = 1, q = 1, η = 0.01, ζ = 0.2.

Comparative StaticsParameter Unlevered Levered Value Optimal Size Optimal Severity

Value firm value firm value of tax shield Leverage upon Coupon of debt-overhangv(x, a, 0; p, q) v(x, a, b∗; p, q) (%) (%) default ad b∗ γb/(1− γ)

Baseline 144.5 150.7 4.3% 35.6% 29.9 3.5 3.9

γ = 0.5 160.5 168.8 5.4% 44.0% 41.4 4.8 4.8γ = 0.6 163.4 167.2 2.3% 16.5% 15.9 1.8 2.7

σ = 0.05 144.6 164.5 4.7% 13.8% 17.4 1.6 1.8σ = 0.15 146.8 147.1 0.2% 6.1% 10.2 0.5 0.6

p = 0.5 238.3 241.2 1.2% 5.2% 12.1 0.8 0.9p = 1.5 209.9 222.7 6.1% 54.7% 59.2 7.9 8.9

q = 0.5 134.0 145.0 8.2% 69.7% 100.0 6.5 7.3q = 1.5 218.0 222.7 2.2% 16.1% 14.9 2.4 2.7

τ = 0.05 156.2 158.2 1.3% 24.7% 25.4 2.4 2.7τ = 0.55 212.8 229.3 7.1% 15.7% 13.7 2.9 3.3

The results demonstrate the ambiguous effects of changes in p as leveredfirm value increases when p is both increased and decreased. The leveredvalue is nevertheless monotonically increasing in the resale price q. More im-portantly, given the choice of parameters, increases in the intermediation priceand declines in the resale price are important drivers of leverage, debt-overhangand over-investment in unproductive assets. In addition, reductions in volatil-

21

ity, decreasing the returns to the intermediation technology or raises taxes arealso important in exacerbating the agency problem.

4 Asset Purchases

Financial intermediaries often expand capacity by purchasing assets.3 Ca-pacity expansions increase the value of the firm directly because additionalcapacity boosts operating profit flows. Moreover, the value of the firm’s op-tions to sell assets become more valuable as sales are no longer irreversible.Nevertheless, assets are not completely reversible because the asset market isassumed to be illiquid, that is Q > q. This illiquidity may arise for exam-ple from informational asymmetries between the firm and outside buyers andsellers. Varying degrees of illiquidity are common in markets for assets tradedby financial intermediaries, especially in over-the-counter (OTC) markets. Thedegree of illiquidity in the asset market as measured by Q−q, is then a measureof the degree to which a firm’s capacity changes are reversible.

4.1 Unlevered Firm

Let Lt denote the cumulative gross investment (i.e. purchases of assets) upto date t. The stochastic process for assets is then:

dat = dLt − dut. (38)

The unlevered firm’s objective is to maximize the expected discounted profitsor

vLu (x, a; p, q,Q) =

maxT∈T ,Lt,ut,t≥0

Ex[∫ ∞

0e−(ρ+η)t π(xt, at; p)dt−QdLt + qdut

](39)

where Ltt≥0 and utt≥0 are nondecreasing continuous stochastic processesthat describe the investment decisions of the firm with L0 = u0 = 0.

The value of the firm now derives from four sources: the value of operatingthe firm’s assets forever, the values associated with the ability to either expandor contract capacity and finally the value associated with the option to abandonand liquidate the firm.

3Evidence collected by Adrian and Shin (2010) points to FIs altering their asset holdings throughrepurchase agreements. I abstract from the contractual form used to acquire additional assets andassume that FIs purchase them directly.

22

The value derived from operating the firm’s assets forever is again the valueof the stream of discounted profit flows from operations:

Π(x, a; p) = (1− τ)Ex[∫ ∞

0e−(ρ+η)t π(xt, at; p)dt

]The option to abandon trades off the value of the assets in place against theliquidation value of the firm. As in Section 3.1, we can write the value of thisoption as [

qaLu −Π(xLu , aLu ; p)

](x/xLu )−λ. (40)

where the abandonment threshold is now xLu and the new the firm size uponabandonment is aLu .

The key difficulty arises in deducing the value of the options to alter capac-ity. This is because changes in capacity are no longer permanent. For example,consider determining the value of the option to reduce capacity. Since capacityreductions are no longer permanent because assets may be purchased later, thevalue of capacity reductions can no longer be written as the difference betweenrevenues and permanent changes in future profit flows as in (17). In fact, thevalue of the two capacity altering options must be determined jointly becausethe ability to expand capacity reduces the irreversibility associated with assetsales and vice-versa.

The optimal investment decisions are now characterized by a pair of thresh-olds xL(a; p, q,Q) > xl(a; p, q,Q) such that investment occurs whenever assetproductivity exceeds the threshold xL(a; p, q,Q) while disinvestment occurswhenever returns are below xl(a; p, q,Q). When asset productivity is betweenxL(a; p, q,Q) and xl(a; p, q,Q) no investment is undertaken.

In order to value the capacity alteration options, I use an approach thatrelies on the HJB equation. It can be shown that the value of capacity optionscan be written as:

α(a;xL, xl)x−λ + β(a;xL, xl)x

θ

where α(a) and β(a) are functions of initial asset level a and the optimal thresh-olds xL and xl, and −λ, and θ are solutions to the fundamental quadratic. Thefunctions α(·) and β(·) can then be determined in terms of the thresholds usingthe following smooth-pasting conditions:

∂vLu∂a

∣∣∣x=xl(a;p,q,Q)

= q when a ∈ [au, a] (41)

∂vLu∂a

∣∣∣x=xL(a;p,q,Q)

= Q when a ∈ [au, a] (42)

23

Furthermore, the optimal thresholds can be determined using the followingsuper-contact conditions:

∂2vLu∂x∂a

∣∣∣x=xl(a;p,q,Q)

= 0 when a ∈ [au, a] (43)

∂2vLu∂x∂a

∣∣∣x=xL(a;p,q,Q)

= 0 when a ∈ [au, a] (44)

While it can be shown that the thresholds xL and xl are continuous and increas-ing functions, closed-form expressions for the thresholds cannot be obtained.

The optimal investment and abandonment decisions in the unlevered caseare depicted in the figure below. As is made clear in the figure, allowing the firm

a

x

XL(a;p,q,Q)

Xu(a;p,q,Q)

au

Xu(au;p,q,Q)

Sell Assets

Do NothingXl(a;p,q,Q)

Abandon

Purchase Assets

Abandon

Abandon

Figure 4: Optimal policies in the unlevered case.

to expand capacity narrows the inaction region. In fact, the inaction regionshrinks as the asset market becomes more liquid, that is as Q − q decreases.In this sense, one can interpret the earlier results without capacity expansionsas the case where Q is very large, yielding a large inaction region.

In addition, as capacity contractions are more valuable in the presence ofthe capacity expansion option, the threshold xl(a; p, q,Q) lies above the corre-sponding threshold xl(a; p, q). As a result, the firm size upon abandonment is

24

smaller.The following proposition provides expressions for the value of the options

and the value of the unlevered firm.

Proposition 4. Assume (ρ + η) > µx > 0. Let −λ, θ be the negative andpositive roots of the fundamental quadratic

(ρ+ η)− µxϑ−1

2σ2(ϑ− 1)ϑ = 0. (45)

Then, the value of the unlevered firm is:

vLu (x, a; p, q,Q) = Π(x, a; p) + α(a;xL, xl)x−λ + β(a;xL, xl)x

θ

+[qaLu −Π(xLu , a

Lu ; p)

](x/xLu )−λ (46)

where

α(a;xL, xl) =

∫ aLu

a

[x−θl (q − πa(xl, a; p)) + x−θL (πa(xL, a; p)−Q)

x−(θ+λ)l − x−(θ+λ)

L

]da (47)

β(a;xL, xl) = −∫ a

aLu

[xλl (q − πa(xl, a; p)) + xλL (πa(xL, a; p)−Q)

xθ+λL − xθ+λl

]da (48)

and the optimal thresholds xL(a; p, q,Q) and xl(a; p, q,Q) are implicitly definedby the following system of equations:

λ

[x−θl (q − πa(xl, a; p)) + x−θL (πa(xL, a; p)−Q))

xλ+1L (x

−(θ+λ)l − x−(θ+λ)

L )

]

− θ

[xλl (q − πa(xl, a; p)) + xλL (πa(xL, a; p)−Q))

x1−θL (xθ+λL − xθ+λl )

]= πxa(xL, a; p) (49)

λ

[x−θl (q − πa(xl, a; p)) + x−θL (πa(xL, a; p)−Q))

xλ+1l (x

−(θ+λ)l − x−(θ+λ)

L )

]

− θ

[xλl (q − πa(xl, a; p)) + xλL (πa(xL, a; p)−Q))

x1−θl (xθ+λL − xθ+λl )

]= πxa(xl, a; p). (50)

Proof. See Appendix A.

In the above expression, α(a;xL, xl)x−λ represents the value of the option to

contract capacity whereas β(a;xL, xl)xθ represent the value of the option to

25

expand capacity. The expressions for α and β above underscore the inter-dependence of the values of the two capacity altering options. Each is anappropriately weighted sum of the marginal benefits associated with sellingassets, (q− πa(xl, a; p)) and purchasing assets, (πa(xL, a; p)−Q). In brief, thevalue of the option to contract is now higher because of the ability to expandcapacity in the future. Similarly, the value of the option to expand capacity ishigher because of the ability to contract capacity at a later point in time.

The optimal abandonment threshold is found as before using the followingsmooth-pasting condition:

∂vLu∂x

∣∣∣x=xLu

= 0.

which gives the following implicit equation for the abandonment threshold xLu :

Πx(xLu , aLu ; p)− λα(aLu , x

Lu , x

Lu )(xLu)−λ−1

+ θβ(aLu , xLu , x

Lu )(xLu)θ−1

− λ

xLu[qaLu −Πx(xLu , a

Lu ; p)] = 0 (51)

However, since the capacity altering options are worth zero upon exit, theabove equation can be solved to yield the following expression for the defaultthreshold:

xLu (a; p, q) =λ((ρ+ η)− µx)

[(1− τ)(c/(ρ+ η) + qaLu

]aγ−1

(1 + λ)(1− τ)p(52)

The firm is abandoned when it is preferable to liquidate the assets rather reducecapacity further. Therefore, in keeping with the last section, the abandonmentsize is determined at the intersection of the thresholds xLu and xl. However,in contrast with the last section, the threshold xl also depends on the optionvalue of future capacity expansions. Therefore, the liquidation decision fullytakes into account the value from the option to expand future capacity.

The comparative statics of the model are summarized in Table 4.1. Thebase parameters were chosen as follows: γ = 0.53, a = 100, x = 1, ρ = 0.05, η =0.01, σx = 0.1, τ = 0.15, c = 1, p = 1, q = 1, Q = 1.1, η = 0.01.. The value ofthe firm is certainly higher when capacity can be increased. The level ofincrease depends of course upon the degree of illiquidity in the asset market.In Table 4.1, we see an increase in firm value on the order of 16% to 0.8% asthe purchase price is raised.

One important source for this increase in value is that capacity reductionsare now more valuable. This can be seen in the decline in firm size uponabandonment as capacity is reduced more aggressively when asset productivityfalls because these reductions are partially reversible.

26

Similarly, changes in q affect not only the value of reducing capacity butalso the value of increasing capacity. In Table 4.1 this is most easily seen asq is increased from 1 to 1.05. The increase in value over and above the casewhen capacity expansions are not possible is roughly 20%.

Comparative StaticsParameter Value Value Value of Size

Value with contraction with contraction expansion uponoption only and expansion option (%) default au

Baseline 144.5 168.2 16.4% 9.7

γ = 0.5 160.5 190.5 18.7% 11.0γ = 0.6 163.4 203.4 24.5% 7.1

p = 0.5 238.3 322.5 35.3% 9.7p = 1.5 209.9 273.6 30.3% 9.7

q = 0.5 134.0 142.0 6.0% 25.1q = 1.05 147.5 175.7 19.1% 8.7

Q = 1.5 144.5 155.1 7.3% 11.2Q = 2.0 144.5 150.0 3.8% 11.8Q = 4.0 144.5 145.6 0.8% 12.4

τ = 0.05 156.2 183.2 17.3% 10.9τ = 0.55 212.8 282.4 32.7% 4.6

4.2 Levered Firm

As in Section 3.2, shareholders in the levered firm make abandonmentand investment decisions to maximize the expected value of the stream ofdiscounted profits:

eLu (x, a, b; p, q,Q)

= maxT∈T ,Lt,ut,t≥0

Ex[∫ ∞

0e−(ρ+η)t [π(xt, at; p)− b]dt−QdLt + qdut

]Again, the cost of servicing debt reduces the profit flow available to share-holders. On the other hand, the ability to expand capacity raises the value

27

of equity. This problem is similar to one facing shareholders in the unleveredcase above, and is solved using the same methods. The following propositioncharacterizes the optimal decisions made by shareholders and the value of thelevered firm.

Proposition 5. Assume (ρ + η) > µx > 0. Let −λ, θ be the negative andpositive roots of the fundamental quadratic

(ρ+ η)− µxϑ−1

2σ2(ϑ− 1)ϑ = 0. (53)

Then, the value of equity in the levered firm is:

eLu (x, a, b; p, q,Q) = Π(x, a, b; p) + α(a, b;xL, xl, b)x−λ + β(a, b;xL, xl, b)x

θ

−Π(xLd , aLd , b; p)(x/x

Ld )−λ (54)

where

α(a, b;xL, xl) =

∫ aLd

a

[x−θl (q − πa(xl, a, b; p)) + x−θL (πa(xL, a, b; p)−Q)

x−(θ+λ)l − x−(θ+λ)

L

]da

(55)

β(a, b;xL, xl) = −∫ a

aLd

[xλl (q − πa(xl, a, b; p)) + xλL (πa(xL, a, b; p)−Q)

xθ+λL − xθ+λl

]da

(56)

and the optimal thresholds xL(a, b; p, q,Q) and xl(a, b; p, q,Q) are implicitlydefined by the following system of equations:

λ

[x−θl (q − πa(xl, a, b; p)) + x−θL (πa(xL, a, b; p)−Q))

xλ+1L (x

−(θ+λ)l − x−(θ+λ)

L )

]

− θ

[xλl (q − πa(xl, a, b; p)) + xλL (πa(xL, a, b; p)−Q))

x1−θL (xθ+λL − xθ+λl )

]= πxa(xL, a, b; p)

(57)

28

λ

[x−θl (q − πa(xl, a, b; p)) + x−θL (πa(xL, a, b; p)−Q))

xλ+1l (x

−(θ+λ)l − x−(θ+λ)

L )

]

− θ

[xλl (q − πa(xl, a, b; p)) + xλL (πa(xL, a, b; p)−Q))

x1−θl (xθ+λL − xθ+λl )

]= πxa(xl, a, b; p).

(58)

Proof. See Appendix A.

The value of equity is simply the value of the discounted profit flow accruingto shareholders plus the value of the capacity altering and default options. Thisdiffers from the unlevered firm value in that the default threshold xLd differsfrom the abandonment threshold xLu , and in that shareholders are not entitledto the liquidation value qaLd .

Moreover, the purchasing and selling thresholds xL and xl are unalteredby debt, as instantaneous profits π are linear in debt so that πa and πxa areactually independent of b. The optimal abandonment threshold is found asbefore using the following smooth-pasting condition:

∂vL

∂x

∣∣∣x=xLd

= 0.

Noting that the capacity altering options are worthless upon default, the aboveequation can be solved for the default threshold:

xLd (a, b; p, q) =λaγ−1((ρ+ η)− µx)(c+ b)

p(ρ+ η)(1 + λ). (59)

The shareholders default when it is preferable to liquidate the assets ratherthan reduce capacity further. Debt shifts the default threshold towards theright, increasing the probability of default. However, recall that with capacityexpansions shareholders reduce capacity more aggressively in response to de-clines in productivity. This is reflected in a higher selling threshold xl. Hence,firm size upon default is higher than in the unlevered case but lower than inthe case without capacity expansions.

The conflict between shareholders and bondholders results nevertheless re-sults in early exit due to debt-overhang and over-investment in unproductiveassets. However, these problems are mitigated with capacity expansions be-cause they reduce the probability of default and reduce firm size upon default.

The optimal decisions, along with the debt-overhang problem are shown inthe figure below:

29

a

x

XL(a,b;p,q,Q)

Xu(a;p,q,Q)

ad

Xd(ad, b;p,q,Q) Sell Assets

Do Nothing

AbandonXd(a,b;p,q,Q)

Purchase Assets

Xl(a,b;p,q,Q)Abandon

Abandon

Figure 5: Optimal policies chosen by shareholders.

The value of debt is simply:

dL(x, a, b; p, q,Q) = Ex[∫ Txd

0e−(ρ+η)tbdt

]+ (1− ζ)qaLdE

x[e−(ρ+η)Txd ] (60)

=b

(ρ+ η)

(1−

(x

xLd

)−λ)+ (1− ζ)qaLd

(x

xLd

)−λ(61)

It is essentially the discounted value of the coupon b/(ρ + η) till default plusthe liquidation value discounted appropriately.

The value of the levered firm can then again be written simply as the sumof the values of equity and debt:

30

Proposition 6. The value of the firm is:

vL(x, a, b; p, q,Q)

= Π(x, a, b; p)−Π(xLd , aLd , b; p)(x/x

Ld )−λ

+ α(a;xL, xl, b)x−λ + β(a;xL, xl, b)x

θ

+b

(ρ+ η)

(1−

(x

xLd

)−λ)+ (1− ζ)qaLd

(x

xLd

)−λ(62)

Table 4.2 summarizes the comparative statics. The base parameters werechosen as follows: γ = 0.53, a = 100, x = 1, ρ = 0.05, η = 0.01, σx = 0.1, τ =0.15, c = 1, p = 1, q = 1, Q = 1.1, η = 0.01, ζ = 0.2.

Comparative StaticsParameter Unlevered Levered Optimal Size Optimal

Value firm value firm value Leverage upon Couponv(x, a, 0; p, q) v(x, a, b∗; p, q) (%) default ad b∗

Baseline 168.2 178.4 22.2% 21.1 2.0

γ = 0.5 162.1 165.8 27.8% 28.4 2.84γ = 0.6 163.9 165.6 9.9% 11.9 1.01

p = 0.5 238.6 239.7 3.4% 10.3 0.5p = 1.5 213.7 218.5 36.4% 41.0 4.9

q = 0.5 134.5 140.4 51.8% 76.8 4.5q = 1.5 222.2 224.3 7.8% 9.5 1.1

Q = 1.5 145.3 148.2 22.8% 21.5 2.1Q = 2.0 146.0 148.7 21.4% 20.5 2.0Q = 4.0 145.3 148.2 22.8% 21.5 2.1

τ = 0.05 157.6 159.1 2.6% 7.7 0τ = 0.55 213.3 228.1 15.8% 13.8 2.8

Again, these results largely parallel those obtained in the last section. How-ever, allowing the shareholders to expand capacity raises the value of equity.In addition, the ability to expand capacity also boosts the value of debt asthe higher capacity generates additional profit flow that can be use to service

31

debt payments. In the simulations shown above, the rise in debt value is notcompletely mitigated by the rise in the value of equity and thus higher leverageis generally exhibited. In addition, the increase in the value of equity leadsshareholder to exit later, mitigating debt-overhang, as they are more willingto inject new capital into the firm.

In the following section, in the industry equilibrium, the resale prices q andQ are determined by the buying and selling behaviour of surviving firms butalso by liquidation of assets from defaulting firms, and liquidations from firmsexiting exogenously or fire-sales. As a result, the industry will be more highlyleveraged and will exhibit greater over-investment in unproductive assets.

5 Industry Equilibrium

In this section, I show that a stationary industry equilibrium exits. Also,I provide intuition for the key steps in the proof while relegating the detailsto the appendix. Finally, I draw on a set of comparative statics to explain thekey implications of the model.

Proposition 7. Assume that

1. (ρ+ η)− µx > 0

2. λ > γ

3. η > σ2x − µx

4. µx − σ2x/2 > 0

where −λ is the negative root of the fundamental quadratic. Then, there existsa stationary equilibrium (p∗, q∗, b∗, xl, xd, N

∗, ν∗), such that x > xd and a > ad.

Proof. See Appendix A

The first assumption above is required to bound the payoffs of the firm. Thesecond is required to bound the higher moments of the stationary distributionsas they have infinite support. Finally, the last two assumptions are needed toensure the existence of the stationary distributions.

I briefly outline the intuition behind the construction of the stationaryequilibrium. Shareholders first select the coupon b given industry prices pand q prior to entry. Since all firms are ex-ante identical they choose anidentical coupon. The relation between the equilibrium prices p∗ and q∗ is thendetermined from the free entry condition. This condition can be understoodas follows. Given a resale price q, whenever the intermediation price is abovethe equilibrium price p∗, there is a positive benefit to entry and thus firms

32

enter. However, entry drives down the intermediation price. Similarly, givenan intermediation price p, whenever the resale price is above q∗, there is againa positive benefit to entry and so firms enter. However, entry drives down theresale price q∗ as some firms that enter wish to reduce capacity upon entry.Analogously, when prices are below equilibrium levels, the value to entry fallsbelow the entry cost ce and firms choose not to enter. Therefore, in order forthere to be positive and finite entry in equilibrium, the value to entering mustequal the entry cost ce.

I then solve compute the optimal selling and default thresholds, xl and xdin terms of the optimal coupon b∗ and equilibrium output prices p∗, q∗. I thencompute the invariant distribution ν∗ up to a scale factor that is the entryrate. The derivations make use of a conditions that matches the incomingflows and outgoing flows of firms and assets in terms of the density of thestationary distribution. This condition can then be solved for the distributionby adapting the procedure outlined in Miao (2005).

Finally, the market clearing conditions are used to pin down the entry rateand the equilibrium prices.

5.1 Results

Comparative StaticsParameter Industry Intermediation Resale Average Size

Value Intermediation Price Price Leverage upon(output) p q (%) default ad

Baseline 164 1.00 1.00 20.1% 25.3

γ = 0.5 156 0.94 0.95 23.5% 26.4γ = 0.6 160 1.07 1.04 19.4% 23.2

σ = 0.05 153 0.95 0.96 36.2% 36.4σ = 0.2 161 1.04 1.06 9.7% 19.2

ζ = 0.1 166 1.02 1.01 21.2% 24.6ζ = 0.3 161 0.98 0.99 19.4% 25.9

τ = 0.10 178 0.93 0.95 18.2% 21.4τ = 0.20 152 1.08 1.04 25.6% 28.6

η = 0.02 178 0.93 0.95 18.2% 21.4η = 0.04 152 1.08 1.04 25.6% 28.6

33

These results are intuitive to interpret. The key set of results are thecomparative statics with respect to the exogenous probability of default η.These suggest that as the possibility of a fire-sale is raised, industry leveragedecreases. This occurs becomes bondholders anticipate a decline in the valueof firm and extend credit on less generous terms. This effect dominates thedecline in equity from the reduced expected cash-flows. While not shown intable above, this result appears to be robust to further changes in η.

In terms of regulatory measures, minimum capital requirements have es-sentially operate as follows. They place a floor on the value of equity thatmust be maintained. Given that the shareholders have the option to abandonthe firm, this type of regulation simply alters their choice of default policy infavour of abandoning earlier. Again, in anticipation of such behaviour, lenderstighten the terms of the debt contract, leading to lower leverage. Nevertheless,clearly minimum capital requirements to not work as intended as they serveto increase the probability of default, rather than decreasing it.

Reducing the ability of firms to sell assets when solvent would possibly bemore effective. Limiting the quantity of assets on the market, limits the pricefeedback effect triggered by the liquidation of defaulting firms. This wouldresult in ex-ante lower value for the firm as the value of the option to reducecapacity is devalued. This would again lead to lower leverage levels, howeverthis time the increase in default rates is more limited. Indeed, only those firmsthat are on the marginally solvent and are selling to service debt would beaffected.

6 Conclusion

This paper investigated the optimal capital structure and default decisionsof firms in an industry competing in both product and asset markets, with thepossibility of fire-sales in the asset market. Fire-sales impact firm financingthrough two channels. One is a direct channel as creditors take into accountthe possibility that each firm may need to resort to a fire-sale of its assets. Theother is through a negative price externality that liquidating firms impose onother firms simply seeking to reduce capacity. Naturally, as the possibility offire-sales is reduced, industry leverage increases while default rates fall.

As for regulation, imposing minimum capital requirements reduces the ex-ante leverage of firms but at the cost of inducing higher default rates. Restric-tions on asset sales reduce the value of the options to increase firm capacity,thereby reducing the value of firm. This reduction in value leads bondholdersto raise borrowing costs anticipating a more severe debt-overhang problem.

34

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A Proofs

Proof of Proposition 1. The proof follows Miao (2005). The value of theunlevered firm is the sum of the expected discounted profits from operatingthe assets forever plus the value of the option to abandon and the value of theoption to default:

vu(x, a; p, q) = maxT∈T ,ut,t∈[0,T ]

(1− τ)Ex[∫ T

0e−(ρ+η)t π(xt, at; p)dt+ qdut

]= Π(x, a; p) + qauEx[e−(ρ+η)Txu ]− (1− τ)Exu

[∫ Txu

0e−(ρ+η)tπ(xt, at; p)dt

]+

∫ au

a

(qEx[e−(ρ+η)Txl(a;p,q) ]− (1− τ)Ex

[∫ ∞Txl(a;p,q)

e−(ρ+η)tπa(xt, at; p)dt

]da

)

The firm term on the second line is:

Π(x, a; p) = (1− τ)Ex[∫ ∞

0e−(ρ+η)t π(xt, at; p)dt

]= (1− τ)

(pxa1−γ

(ρ+ η)− µx− c

ρ+ η

)where the second line follows from computing the expectation directly. Usingthe strong-Markov property of stopping times of Brownian motion, the second-term can be written as:

qauEx[e−(ρ+η)Txu ]− Ex[e−(ρ+η)Txu ]

[∫ ∞0

e−(ρ+η)tπ(xt, at; p)dt

]= [qau −Π(xu, au; p)] (x/xu)−λ

where the second line following from noting that Ex[e−(ρ+η)Txu ] = (x/xu)−λ

(see Karatzas and Shreve (1991), p. 191). Combining the above points, thefinal term can be written as:∫ au

a

(q(x/xl)

−λ − (1− τ)Ex[e−(ρ+η)Txl ]Ex[∫ ∞

0e−(ρ+η)tπa(xt, at; p)dt

]da

)=

∫ au

a

(q(x/xl)

−λ − (1− τ)Ex[e−(ρ+η)Txl ]Ex[∫ ∞

0e−(ρ+η)tπa(xt, at; p)dt

]da

)=

∫ au

a(x/xl)

−λ(q − (1− τ)Ex

[∫ ∞0

e−(ρ+η)tπa(xt, at; p)dt

]da

)=

∫ au

a(x/xl)

−λ(q − (1− τ)

[(1− γ)pxa−γ

(ρ+ η)− µx

])da

38

Then, the optimal threshold (23) is found using the super-contact conditon(see Dumas (1991)):

∂2vu∂x∂a

∣∣∣x=xl(a;p,q)

= 0

and the optimal abandonment threshold (24) is found using the smooth-pastingcondition (see Dixit and Pindyck (1994)):

∂vu∂x

∣∣∣x=xu

= 0.

Then, using (23), and (24) in the expressions above yields (22). Finally, notingthat the above thresholds meet at au, equating (23), and (24) and solving forau yields (25).

Proof of Proposition 2. The expression for the value of equity is:

e(x, a, b; p, q)

= maxT∈T ,ut,t∈[0,T ]

(1− τ)Ex[∫ T

0e−(ρ+η)t [π(xt, at; p)− b]dt+ qdut

]which is identical to the expression for the unlevered value minus the liquida-tion value. As a result, the method of proof is identical to the method used inproving Proposition 1. Applying the super-contact and smooth-pasting condi-tions, yields (31) and (32). The firm size at abandonment can again be foundby equating (31) and (32) at ad. As in Morellec (2001), existence and unique-ness follow from the fact that ∂xl/∂a > ∂xd/∂a, and for a > ad, xl > xd.

Proof of Corollary 1. The expressions for au and ad are given respectivelyby (25) and (33). Then, ad > au when:

(c+ b)(1− γ)(1− τ)

q(ρ+ η)>c(1− γ)(1− τ)

q(ρ+ η)γ(63)

=⇒ bγ

1− γ> c (64)

Proof of Proposition 3. The expression (36) follows from adding the ex-pressions for equity (30) and debt (34).

39

Proof of Proposition 4. Then, the value of the unlevered firm is:

vLu (x, a; p, q,Q) = Π(x, a; p) + α(a;xL, xl)x−λ + β(a;xL, xl)x

θ

+[qaLu −Π(xLu , a

Lu ; p)

](x/xLu )−λ

To write the above expression in terms of the thresholds xl and xL we use thesmooth-pasting conditions:

∂vLu∂a

∣∣∣x=xl(a;p,q,Q)

= q when a ∈ [au, a]

∂vLu∂a

∣∣∣x=xL(a;p,q,Q)

= Q when a ∈ [au, a]

These yield:

α′x−λ + β′xθ +∂Π(x, a; p)

∂a

∣∣∣x=xL

= Q (65)

α′x−λ + β′xθ +∂Π(x, a; p)

∂a

∣∣∣x=xl

= q. (66)

These equations can be solved for α′ and β′ in terms of the thresholds xl andxL to give (47) and (48). The optimal thresholds can be determined using thefollowing super-contact conditions:

∂2vLu∂x∂a

∣∣∣x=xl(a;p,q,Q)

= 0 when a ∈ [au, a]

∂2vLu∂x∂a

∣∣∣x=xL(a;p,q,Q)

= 0 when a ∈ [au, a]

which yield:

−λα′x−λ + θβ′xθ +∂Π(x, a; p)

∂a∂x

∣∣∣x=xL

= 0 (67)

−λα′x−λ + θβ′xθ +∂Π(x, a; p)

∂a∂x

∣∣∣x=xl

= 0. (68)

Replacing α′ and β′ in the equations above by the expressions in (47) and (48),we obtain the two equations (49) and (50) that define the optimal thresholds.

40

Proof of Proposition 5. The expression for equity is:

eLu (x, a, b; p, q,Q) = Π(x, a, b; p)−Π(xLd , aLd , b; p)(x/x

Ld )−λ

+ α(a;xL, xl, b)x−λ + β(a;xL, xl, b)x

θ (69)

which is similar to the expression for the value of the firm in Proposition 4.The expression for equity (54), as well as the optimal thresholds, are foundusing the same method as in Proposition 4.

Proof of Proposition 6. The expression (62) follows from adding the ex-pressions for equity (54) and debt (60).

Proof of Proposition 7. The proof follows Dixit and Pindyck (1994) andMiao (2005). The first step consists of deriving a condition linking the inter-mediation price p∗ and the resale price q∗ using the free-entry condition. Thecapacity reduction and default thresholds can then be found from Proposition2 above. Then, the equilibrium distribution of firms ν∗ is derived. Finally, theequilibrium prices can be determined using the market clearing conditions.

To show the existence of a stationary distribution, it is convenient to workwith the logarithm of x. Let z = log x. Then, ztt≥0 is a Brownian motionwith growth rate µz = µx − 1

2σ2x, and volatility σz = σx. Then, the initial

draw of z = log(x) has an exponential distribution over [z, z] where z = log xand z = log x. This is because the initial draw of x is uniform over [x, x]. Thedensity of this distribution is given by:

g1(z) = exp(z − z) (70)

where z = log(x − x). As shown in Harrison (1985), p90-92, the distributionof assets also follows a geometric Brownian motion with growth rate [µ −12σ

2x]γ. Then, letting a = log a, the above logic tells us that the density of the

distirbution of a is given by:

g2(a) = exp(a− a), (71)

where a = log(a− a), where a has growth rate µa = [µ− 12σ

2x]γ and volatility

σa = σx. As the draws are independent across x and a, we can write thedensity of the joint distribution of z, a as:

g(z, a) = exp(z − z) exp(a− a). (72)

41

Now, denote by N∗φ(z, a) the stationary distribution of incumbent firmswith support on [zd,∞)× [ad,∞), where zd = log xd, ad = log ad and N∗ is theentry rate which will be determined later. Following Dixit and Pindyck (1994),for stationarity the density φ must satisfy the following partial differentialequation:

1

2φ2zφzz − µzφz − ηφ+ g(z, a) = 0 (73)

A particular solution to this equation is:

φ0(z, a) =exp(z − z) exp(a− a)

η + µz − σ2x/2

(74)

The general solution is: A1(a) exp δ1z +A2(a) exp δ2z where A1(a) and B1(a)are functions to be determined using the boundary conditions, and δ1, δ2 areroots of the corresponding fundamental quadratic. The functions differ in thefollowing three cases: z ≤ z < z, zd ≤ z < z, and z ≥ z. These functionscan then be found using an appropriate set of boundary conditions as in Miao(2005).

42